Universal Quantum Criticality in the Metal-Insulator Transition of Two-Dimensional Interacting Dirac Electrons

Abstract

The metal-insulator transition has been a subject of intense research since Nevil Mott has first proposed that the metallic behavior of interacting electrons could turn to the insulating one as electron correlations increase. Here, we consider electrons with massless Dirac-like dispersion in two spatial dimensions, described by the Hubbard models on two geometrically different lattices, and perform numerically exact calculations on unprecedentedly large systems that, combined with a careful finite size scaling analysis, allow us to explore the quantum critical behavior in the vicinity of the interaction-driven metal-insulator transition. We find thereby that the transition is continuous and determine the quantum criticality for the corresponding universality class, which is described in the continuous limit by the Gross-Neveu model, a model extensively studied in quantum field theory. We furthermore discuss a fluctuation-driven scenario for the metal-insulator transition in the interacting Dirac electrons: the metal-insulator transition is triggered only by the vanishing of the quasiparticle weight but not the Dirac Fermi velocity, which instead remains finite near the transition. This important feature cannot be captured by a simple mean-field or Gutzwiller-type approximate picture, but is rather consistent with the low energy behavior of the Gross-Neveu model.

Critical behavior in the presence of an order-parameter pinning field

“…By extrapolating the magnetization to the thermodynamical limit, Ref. [1] located the critical point at U/t = 3.78, a value in line with recent numerical simulations [75,76]. According to the present study, the pinning field is a relevant perturbation for the line critical behavior, so that for h 0 = 0, under the RG the model flows away from the "ordinary" fixed point h 0 = 0.…”

“…By means of MC simulations, we have checked this scenario for both the bilayer Heisenberg model and the improved Blume-Capel model, and verified the exact result for the scaling dimension of the pinning field at the h 0 = 0 fixed point. This picture is also expected to hold for the Hubbard model on the honeycomb lattice, which undergoes a quantum phase transition in the Gross-Neveu Heisenberg UC [74][75][76]. For this model the scaling dimension of the pinning field is found to be significantly smaller than for the O(N) models, such that very large lattice sizes would be needed in order to reach the asymptotic behavior; see the discussion in Appendix B.…”

Extracting the Single-Particle Gap in Carbon Nanotubes with Lattice Quantum Monte Carlo

“…Past numerical simulations of graphene suggest that there is a phase transition between semi-metal and Mott insulator that occurs near U c /κ ≈ 3.5 [16]. More recent calculations have pushed this value larger, U c /κ ≈ 3.8 [12]. Since matching the nanotube data favors a lower U, our results suggest that graphene is a semi-metal.…”

“…Our correlation functions are accurate to O(δ 2 ) [15], and so the calculated effective masses should scale as O(δ), which motivates the use of Eq. (12). We combine our extrapolated points in the right panel of Figure 4 and show the dependence of ∆/κ as a function of Hubbard ratio U/κ for different tube geometries.…”

Self-learning quantum Monte Carlo method in interacting fermion systems

“…DQMC [15][16][17] has been widely used in the investigation of correlated fermion systems [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Despite of the great successes, the method also suffers from serious difficulties.…”